A method for long-time integration of Lyapunov exponent and vectors along fluid particle trajectories

Finite-time Lyapunov exponents (FTLEs) and Lyapunov vectors (LVs) are powerful tools to illustrate Lagrangian coherent structures (LCSs) in experiments and numerical simulations of fluid flows. To obtain the FTLEs and LVs with the flow simulation simultaneously, we computed the eigenvectors and eigenvalues of the left Cauchy-Green tensor along the trajectories of fluid particles separately instead of computing deformation gradient tensor directly. The method proposed in the present study not only avoids solving the eigenvalue problem of the singular matrix at each time step but also guarantees a stable simulation for a long time. The method is applied in the computation of FTLEs and LVs in two-/three-dimensional (2D/3D) compressible/incompressible cases. In 2D cases, we found that LCSs are folded as fine filaments induced by vortices, while LCSs are sheet-like structures among the vortices for 3D cases. Meanwhile, the directions of stretching and compression of LVs are tangent and normal to the FTLE ridges (2D)/iso-surfaces (3D), respectively.

Automated summary

Finite-time Lyapunov exponents (FTLEs) and Lyapunov vectors (LVs) are powerful tools for analyzing Lagrangian coherent structures (LCSs) in fluid flows. A new method is proposed in this study to compute FTLEs and LVs by calculating eigenvectors and eigenvalues of the left Cauchy-Green tensor, avoiding the singularity problem in matrices. In 2D cases, LCSs appear as filaments induced by vortices, while in 3D cases, LCSs are sheet-like structures among the vortices.